From Knot Atlas
Jump to: navigation, search






(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a106 at Knotilus!

Link Presentations

[edit Notes on L10a106's Link Presentations]

Planar diagram presentation X10,1,11,2 X18,11,19,12 X20,5,9,6 X14,7,15,8 X12,4,13,3 X16,14,17,13 X6,15,7,16 X8,9,1,10 X4,19,5,20 X2,18,3,17
Gauss code {1, -10, 5, -9, 3, -7, 4, -8}, {8, -1, 2, -5, 6, -4, 7, -6, 10, -2, 9, -3}
A Braid Representative
A Morse Link Presentation L10a106 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^3 t(1)^3-2 t(2)^2 t(1)^3+t(2) t(1)^3-3 t(2)^3 t(1)^2+7 t(2)^2 t(1)^2-5 t(2) t(1)^2+2 t(1)^2+2 t(2)^3 t(1)-5 t(2)^2 t(1)+7 t(2) t(1)-3 t(1)+t(2)^2-2 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial \frac{13}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{13}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{11}{q^{11/2}}-4 \sqrt{q}+\frac{7}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 \left(-z^3\right)-2 a^7 z+2 a^5 z^5+6 a^5 z^3+5 a^5 z+a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-5 a^3 z-a^3 z^{-1} +a z^5+2 a z^3 (db)
Kauffman polynomial -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-z a^9-5 z^6 a^8+4 z^4 a^8-z^2 a^8-7 z^7 a^7+10 z^5 a^7-8 z^3 a^7+z a^7-6 z^8 a^6+7 z^6 a^6-2 z^4 a^6-z^2 a^6-2 z^9 a^5-10 z^7 a^5+33 z^5 a^5-30 z^3 a^5+10 z a^5-a^5 z^{-1} -11 z^8 a^4+24 z^6 a^4-12 z^4 a^4+z^2 a^4+a^4-2 z^9 a^3-7 z^7 a^3+31 z^5 a^3-27 z^3 a^3+9 z a^3-a^3 z^{-1} -5 z^8 a^2+11 z^6 a^2-3 z^4 a^2-z^2 a^2-4 z^7 a+11 z^5 a-8 z^3 a+z a-z^6+2 z^4-z^2 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
j \
4          1-1
2         3 3
0        41 -3
-2       73  4
-4      75   -2
-6     76    1
-8    67     1
-10   57      -2
-12  27       5
-14 14        -3
-16 2         2
-181          -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.